In many longitudinal studies, repeated response and predictors are not directly observed, but can be treated as distorted by unknown functions of a common confounding covariate. Moreover, longitudinal data involve an observation process which may be informative with a longitudinal response process in practice. To deal with such complex data, we propose a class of flexible semiparametric covariate-adjusted joint models. The new models not only allow for the longitudinal response to be correlated with observation times through latent variables and completely unspecified link functions, but they also characterize distorted longitudinal response and predictors by unknown multiplicative factors depending on time and a confounding covariate.

Theta functions are special functions of two complex variables, which is a central subject in the study of mathematics. Theta functions play a key role in many branches of mathematics, including algebraic geometry, number theory, modular forms, partial differential equations and combinatorics. In this talk, I will give a brief introduction to theta functions with emphasis on theta function identities.

In this talk we discuss some monotonicity questions related to Fredholm matrices and operators. A function $f(x)$ is called completely monotonic if $(-1)^mf^{(m)}(x)>0$. It is known that the expectation of having $m$ eigenvalues of a random Hermitian matrix in an interval is a multiple of $(-1)^{m}$ times the $m$-th derivative of a Fredholm determinant at $/lambda=1$. In this work we extend the positivity to half-real line $(-/infty,1]$, and we also study the completely monotonicity of some special functions which arise as Fredholm determinants.

We survey results on q-analogue of the normal distribution and the corresponding orthogonal polynomials.

In this talk, we will investigate a complex short pulse equation and its two-component generalization. First, we will derive them starting from Maxwell equations. Then we will study their various soliton solutions such as bright, dark, breather and rogue wave solutions by using Hirota’s bilinear method and Darboux transformation method.

We generalize the notions of ordinary points and singularities from linear ordinary differential equations to D-finite systems. Ordinary points and apparent singularities of a D-finite system are characterized in terms of its formal power series solutions. We also show that apparent singularities can be removed like in the univariate case by adding suitable additional solutions to the system at hand. Several algorithms are presented for removing and detecting apparent singularities. In addition, an algorithm is given for computing formal power series solutions of a D-finite system at apparent singularities. This is a joint work with Manuel Kauers, Ziming Li and Yi Zhang.

Log Fano manifold are generalization of Fano manifolds, and the canonical metrics on them are conic Kahler-Einstein metrics. I will talk about the existence of such metrics on log K-polystable log Fano manifolds.

We establish spreading properties of the Lotka-Volterra competition-diffusion system. When the initial data vanish on a right half-line, we show that there are two successive invasions. We derive the formulas of the exact spreading speeds and prove the convergence to homogeneous equilibrium states in between the invasion fronts. Our method is inspired by the geometric optics approach for Fisher-KPP equation due to the work of Freidlin, and that of Evans and Souganidis. Our main result settles an open question raised by Shigesada and Kawasaki in 1997, and shows that one of the species spreads to the right with a speed which is linearly, but non-locally, determined.